The concept of real rank zero of a $C^*$-algebra is introduced as non-commutative analogue of dimension ( topological dimension ). For example, it shown (by Brown-Pedersen) such that, if $X$ is a compact Hausdorff space, then $RR$ $C(X)=dim(X)$.

Let $A$ be a $C^*$-algebra. we define $RR(A)=0$ iff every self-adjoint element in $A$ can be approximate by invertible, self-adjoint element in $A$. for example every Von-Neumman algebra has real rank zero.

So, on the other hand since $C^*(\mathbb T)=C(\mathbb Z)$ and $C^*(\mathbb Z)=C(\mathbb T)$, then $RR$ $C^*(\mathbb T)=0$ but
$RR$ $C^*(\mathbb Z)$$\not=$$0$ .

Question: Which locally compact groups $G$ satisfy $RR$ $C^*(G)=0$?

NOTE: If $G$ is locally compact group with fixed Harr measure, let full group $C^*$-algebra $C^*(G)$ be the completion of the convolution algebra $L^1(G)$ with respect to the norm $\lVert f\rVert_{C^*(G)}$=$\sup $$\{\lVert\phi(f)\rVert\}$, where the supremum is taken over all $*$-representation $\phi$ of $L^1(G)$ as an algebra of bounded linear operators in a Hilbert space.

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    $\begingroup$ BTW, I find this a much better question than your older one about "ideal structure", since you have a more limited and hence more focused question. $\endgroup$ – Yemon Choi Feb 14 '16 at 19:11

Here is a partial answer, and a pointer towards some literature that should be relevant.$\newcommand{\Cst}{{\rm C}^*}$

Doing some searching online brings up an old result of Kaniuth (Proc. AMS, 1993), which implies that if $G$ is a connected non-compact group then its full $\Cst$-algebra has real rank strictly greater than zero. The proof relies heavily on structure theory for Lie groups, but the structure theory used seems to be fairly standard material from that area.

Theorem 2 of Kaniuth's paper gives some examples of nilpotent locally compact groups whose $\Cst$-algebras have real rank zero.

Note that the $c_0$-direct sum of finite-dimensional matrix algebras always has real rank zero (this is a nice exercise in understanding the definition) and hence $\Cst(G)$ has real rank zero for every compact group $G$.


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